⚛️ quantum physics

Reconstructing non-Abelian braiding and fusion without anyon transport

This paper presents and experimentally demonstrates a measurement-only protocol on Quantinuum's H2 trapped-ion processor that successfully reconstructs the non-Abelian braiding and fusion primitives of the $D(S_3)$ quantum double model with high fidelity, eliminating the need for physical anyon transport.

Original authors: Lucy Byles, Matthew D. Horner, Benjamin T. H. Varcoe, Jiannis K. Pachos

Published 2026-08-06

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Lucy Byles, Matthew D. Horner, Benjamin T. H. Varcoe, Jiannis K. Pachos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe is built from tiny, invisible Lego bricks. Most of us know these bricks come in two flavors: the smooth, round ones that like to pile up neatly (bosons) and the shy, antisocial ones that refuse to sit next to each other (fermions). But in a strange, exotic corner of physics called "topological matter," there are third-party bricks called anyons. These aren't just particles; they are more like knots in a string. If you swap two normal particles, nothing much happens. But if you swap two anyons, they leave a permanent "memory" of their dance, changing the state of the universe in a way that depends entirely on the order of their moves.

This memory is the holy grail for a new kind of computer. Because the information is stored in the shape of the dance rather than the position of a single brick, it's incredibly hard to mess up. A little bump or a stray breeze (what scientists call "noise") can't easily untie the knot. This is the promise of topological quantum computing: a machine that is naturally immune to errors. However, there's a catch. To make these computers work, we usually need to physically move these anyon-knots around each other, like weaving a basket. In the real world, dragging these particles around is incredibly hard, slow, and prone to breaking the very knots we're trying to protect.

So, the big question for physicists is: Can we get the benefits of this magical knot-weaving without actually having to drag the knots across the table?

This paper says, "Yes, we can." A team of researchers from the University of Leeds and Aegiq Ltd. has successfully demonstrated a clever trick to simulate the complex dance of these anyons without ever physically moving them. Instead of dragging particles around a circuit board, they used a "measurement-only" approach. Think of it like this: usually, to see how two people interact, you watch them walk past each other. But here, the researchers decided to just ask a series of very specific questions about their relationship at different times. By carefully ordering these questions (measurements), they could reconstruct the exact same "dance moves" and "knot patterns" that would have happened if they had physically moved the particles.

The team focused on a specific, tricky type of anyon model based on a mathematical group called $S_3$ . In the real world, this model requires a lot of complex machinery to describe. The researchers realized they could shrink this massive, complicated system down to a tiny, manageable version involving just two "qutrits" (a three-level version of a computer bit). They then ran this mini-protocol on Quantinuum's H2 trapped-ion quantum processor, a super-powerful machine that uses suspended atoms as its bits.

The results were strikingly accurate. By using a technique called an "adapted Hadamard test" (a fancy way of peeking at the quantum state) and carefully selecting only the successful outcomes, they reconstructed the "braiding" (the dance) and "fusion" (the merging) rules of these anyons. When they checked their work, the reconstructed dance moves matched the perfect, theoretical moves with an average accuracy of 0.9988 for braiding and 0.9987 for fusion. To put that in perspective, if you were trying to copy a perfect circle, you'd be off by less than a hair's width.

Crucially, they didn't just copy the moves; they proved these reconstructed moves could create something special. By combining their reconstructed braiding and fusion steps, they generated a "non-Clifford" resource state. In the language of quantum computing, this is like finding a new color that doesn't exist in the standard paint set. It's a necessary ingredient for building a truly universal quantum computer that can solve any problem, not just the easy ones.

The paper explicitly rules out the idea that you must physically transport anyons to get these results. They showed that the "transport" part is just a story we tell to understand the math; the real magic happens through the timing of measurements. While the experiment was a success, the authors are careful to note that the fusion part of their experiment required a lot of "post-selection," meaning they had to throw away most of their data runs because they didn't meet strict criteria. This made the fusion results a bit noisier than the braiding results, but the overall fidelity remained incredibly high.

In short, this work proves that we can build the foundation for a fault-tolerant quantum computer by using a "virtual" dance floor. We don't need to drag the particles around; we just need to ask the right questions in the right order. This opens the door to modular, scalable quantum processors where the complex topological magic is handled by software and measurement, rather than the impossible task of physically shuffling fragile quantum knots around a chip.

=== DRAFT ===
Here is a detailed technical summary of the paper "Reconstructing non-Abelian braiding and fusion without anyon transport."

Problem Statement

Non-Abelian anyons offer a pathway to fault-tolerant and universal quantum computing due to their ability to store logical information non-locally, rendering it insensitive to local perturbations. However, experimentally accessing the defining data of these anyons—specifically the braiding ( $R$ ) and fusion ( $F$ ) transformations—remains a significant challenge. Conventional protocols require the controlled creation, physical transport, and coherent measurement of anyonic excitations, which incurs substantial resource overhead on quantum hardware. While "measurement-only" topological quantum computation has theoretically demonstrated that braiding can be generated via temporally ordered charge measurements rather than physical motion, directly reconstructing the underlying $R$ and $F$ data for non-Abelian models (such as the quantum double $D(S_3)$ ) on digital quantum hardware has been hindered by the complexity of the required lattice descriptions, which involve local $d=6$ qudits and non-unitary operations.

Methodology

The authors introduce and experimentally realize a measurement-only protocol that reconstructs the non-Abelian braiding and fusion primitives of the $D(S_3)$ quantum double model without physically transporting anyons. The approach relies on three key methodological innovations:

Dense Reduction: The authors demonstrate that the relevant four-qudit overlaps required to extract $D(S_3)$ data can be reduced to expectation values of operators acting on only two qutrits. By exploiting the group structure of $S_3 \simeq Z_3 \rtimes Z_2$ , the complex four-qudit plaquette dynamics are mapped onto a reduced two-qutrit system. Crucially, these expectation values can be evaluated on a simple product state $|\xi\rangle$ rather than the entangled ground state of the full plaquette, significantly simplifying state preparation.
Qubit Encoding and Ancilla-Assisted Dilation: The reduced two-qutrit protocol is implemented on Quantinuum's H2 trapped-ion processors. Each logical qutrit is encoded in the unit-Hamming-weight subspace of three physical qubits ( $|1,0,0\rangle, |0,1,0\rangle, |0,0,1\rangle$ ). Since the ribbon and charge-projection operators are non-unitary, they are implemented probabilistically using ancilla-assisted dilations. Successful operations are heralded by specific measurement outcomes on ancilla qubits.
Reconstruction Protocols:
- Braiding: The squared braiding phases $(R_i)^2$ are reconstructed using an adapted Hadamard test. This allows for the extraction of both real and imaginary components of the non-unitary ribbon-projection overlaps. The $G$ -sector phase is inferred by subtracting the $+$ -sector result from an identity-channel measurement.
- Fusion: The squared fusion amplitudes $|F_{ij}|^2$ are reconstructed via post-selected overlap and normalization probabilities. The protocol prepares two fusion-basis resolutions of the same anyonic process; the vacuum expectation value of the resulting closed diagram yields the squared magnitude of the fusion matrix elements.

Key Contributions

Experimental Realization of Measurement-Only Protocols: The paper provides a compact, observable-specific protocol that reconstructs non-Abelian braiding and fusion data using temporally ordered ribbon operations and local charge projections, eliminating the need for physical anyon transport. This approach complements previous large-scale realizations (e.g., Ref. [24]) by compressing the relevant plaquette histories into a minimal two-qutrit logical protocol suitable for characterizing elementary primitives on qubit-based hardware.
Compact Two-Qutrit Implementation: By reducing the $D(S_3)$ plaquette dynamics to a two-qutrit logical protocol, the authors demonstrate a modular route to characterizing non-Abelian primitives on qubit-based hardware, bypassing the need for large-scale $d=6$ qudit encodings.
High-Fidelity Primitive Reconstruction: The study successfully reconstructs the squared braiding phases and fusion amplitudes for the minimal non-Abelian quantum double $D(S_3)$ , focusing on the closed fusion subcategory $\{A, B, G\}$ (where $A$ is vacuum, $B$ is Abelian, and $G$ is non-Abelian).

Results

The experiment was conducted on Quantinuum's H2 trapped-ion processors. The results demonstrate high fidelity between the reconstructed transformations and the ideal theoretical values:

Braiding Phases: The reconstructed squared braiding phases $(R_+)^2$ and $(R_G)^2$ were extracted with average normalized output-state fidelities of $F_R = 0.9988$ . The experimental values agreed with the theoretical roots of unity ( $\bar{\omega}$ and $\omega$ ) with statistical significance exceeding five standard deviations from the zero-amplitude depolarization limit.
Fusion Amplitudes: The squared fusion amplitudes were reconstructed from post-selected probabilities. The resulting fusion matrix $F_{exp}$ achieved an average normalized output-state fidelity of $F_F = 0.9987$ when tested against Haar-random input states. The reconstruction of the $G$ -channel entry ( $F_{GG}$ ) was consistent with the theoretical value of zero, as no accepted events were observed in the $m_{GG}$ channel (0 out of 11,000 shots), while the dominant uncertainty in other entries arose from the low acceptance probability of the $G$ -normalization channel ( $p(n_G) \approx 4.6 \times 10^{-4}$ ), which yielded only 6 accepted events out of 13,000 shots.
Non-Clifford Resource Generation: Combining the reconstructed primitives, the authors generated a composite braid $B_s = F_s^\dagger R^2 F_s$ . This operation produced a non-stabilizer resource state, quantified by a second-order stabilizer Rényi entropy $M_2 \approx 0.258 \pm 0.0886$ , distinct from the ideal value of $\approx 0.208$ . This confirms the generation of a non-Clifford resource, a necessary component for universal quantum computation.

Significance and Claims

The paper claims that these results establish a compact hardware building block for programmable non-Abelian topological processing. By compressing the relevant plaquette histories into a two-qutrit logical protocol, the authors provide a modular route to characterizing elementary braiding and fusion primitives with high accuracy before assembling larger structures.

The work supports the viability of "measurement-only" anyonic encodings as building blocks for larger topologically encoded quantum processors. It demonstrates that the underlying $R$ and $F$ data can be directly reconstructed on quantum hardware without the resource overhead of extended topological state preparation or physical anyon transport. The authors note that while the current experiment addresses a single encoded plaquette, the combination of compact logical representation, high-fidelity primitives, and a modular measurement-only architecture offers a promising route toward scaling braiding, fusion, magic-state preparation, and logical gates within larger encoded anyonic processors.

Problem Statement

Methodology

Key Contributions

Results

Significance and Claims

More like this